If you ban breeding altogether, then you will eventually wipe out dogs altogether (which I'm sure PETA would love). There is still a place for responsible and ethical breeders to produce dogs and to keep gene pools alive and healthy. But there does need to be stricter controls to minimise any genetic health conditions and to provide a healthy animal population.
There is still a strong need for purebred animals that were bred for a purpose. The original breeders of the labradoodle, the Guide Dogs, abandoned the project because they were not suitable for purpose and have gone back to purebred labradors, with the outsourcing of other purebreds when the purpose requires (usually pedigree poodles for those with allergies, but I also know a few lagottos being used). Other therapy organisations have the same reasoning because they need to know what they are getting. Other dogs are bred for specific purposes - hunting, herding, search and rescue, police work, sniffer dog work. We need to maintain these, whilst also supporting rescue work, and trying to reduce the number of dogs in rescues.
I've long said that this needs to be from education of the buying public. You would never buy a car without expecting a warranty or a second hand car without expecting it to have a roadworthy check first, so why are people so willing to buy a dog who could be a part of their family for the next 15 years without expecting the parents to have any required health testing for their breed/breeds? Educate people where to responsibly and reliably source their pets and we might start to make a dent in the rescue population.